∫ (4x)(x^2+5)^3 wrt x eching

$$\int ( 4 x ) ( x ^ { 2 } + 5 ) ^ { 3 } d x$$

Baholash

$$\frac{x^{8}}{2}+10x^{6}+75x^{4}+250x^{2}+С$$

Yechim qadamlari

\left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} binom teoremasini \left(x^{2}+5\right)^{3} kengaytirilishi uchun ishlating.

$$\int 4x\left(\left(x^{2}\right)^{3}+15\left(x^{2}\right)^{2}+75x^{2}+125\right)\mathrm{d}x$$

Daraja ko‘rsatkichini boshqa ko‘rsatkichga oshirish uchun, darajalarini ko‘paytiring. 2 va 3 ni ko‘paytirib, 6 ni oling.

$$\int 4x\left(x^{6}+15\left(x^{2}\right)^{2}+75x^{2}+125\right)\mathrm{d}x$$

Daraja ko‘rsatkichini boshqa ko‘rsatkichga oshirish uchun, darajalarini ko‘paytiring. 2 va 2 ni ko‘paytirib, 4 ni oling.

$$\int 4x\left(x^{6}+15x^{4}+75x^{2}+125\right)\mathrm{d}x$$

4x ga x^{6}+15x^{4}+75x^{2}+125 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$\int 4x^{7}+60x^{5}+300x^{3}+500x\mathrm{d}x$$

Summani muddatma-muddat integratsiya qiling.

$$\int 4x^{7}\mathrm{d}x+\int 60x^{5}\mathrm{d}x+\int 300x^{3}\mathrm{d}x+\int 500x\mathrm{d}x$$

Har bir shartda konstantani qavsdan tashqariga oling.

$$4\int x^{7}\mathrm{d}x+60\int x^{5}\mathrm{d}x+300\int x^{3}\mathrm{d}x+500\int x\mathrm{d}x$$

k\neq -1 uchun integral \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} boʻlgani uchun, \int x^{7}\mathrm{d}x integralni \frac{x^{8}}{8} bilan almashtiring. 4 ni \frac{x^{8}}{8} marotabaga ko'paytirish.

$$\frac{x^{8}}{2}+60\int x^{5}\mathrm{d}x+300\int x^{3}\mathrm{d}x+500\int x\mathrm{d}x$$

k\neq -1 uchun integral \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} boʻlgani uchun, \int x^{5}\mathrm{d}x integralni \frac{x^{6}}{6} bilan almashtiring. 60 ni \frac{x^{6}}{6} marotabaga ko'paytirish.

$$\frac{x^{8}}{2}+10x^{6}+300\int x^{3}\mathrm{d}x+500\int x\mathrm{d}x$$

k\neq -1 uchun integral \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} boʻlgani uchun, \int x^{3}\mathrm{d}x integralni \frac{x^{4}}{4} bilan almashtiring. 300 ni \frac{x^{4}}{4} marotabaga ko'paytirish.

$$\frac{x^{8}}{2}+10x^{6}+75x^{4}+500\int x\mathrm{d}x$$

k\neq -1 uchun integral \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} boʻlgani uchun, \int x\mathrm{d}x integralni \frac{x^{2}}{2} bilan almashtiring. 500 ni \frac{x^{2}}{2} marotabaga ko'paytirish.

$$\frac{x^{8}}{2}+10x^{6}+75x^{4}+250x^{2}$$

Агар F\left(x\right)f\left(x\right) ning dastlabki holati boʻlsa, u holatda f\left(x\right) ning barcha dastlabki holatlari toʻplami F\left(x\right)+C tarafidan belgilanadi. Shu sababli natijaga C\in \mathrm{R} integrallash konstantasini qoʻshing.

$$250x^{2}+75x^{4}+10x^{6}+\frac{x^{8}}{2}+С$$

x ga nisbatan hosilani topish

$$4x\left(x^{2}+5\right)^{3}$$